\(\int \frac {x^3 (a+b \text {csch}^{-1}(c x))}{(d+e x^2)^3} \, dx\) [113]

Optimal result

Integrand size = 21, antiderivative size = 167 \[ \int \frac {x^3 \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=-\frac {b c x \sqrt {-1-c^2 x^2}}{8 \left (c^2 d-e\right ) e \sqrt {-c^2 x^2} \left (d+e x^2\right )}+\frac {x^4 \left (a+b \text {csch}^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}+\frac {b c \left (c^2 d-2 e\right ) x \text {arctanh}\left (\frac {\sqrt {e} \sqrt {-1-c^2 x^2}}{\sqrt {c^2 d-e}}\right )}{8 d \left (c^2 d-e\right )^{3/2} e^{3/2} \sqrt {-c^2 x^2}} \] Output:

-1/8*b*c*x*(-c^2*x^2-1)^(1/2)/(c^2*d-e)/e/(-c^2*x^2)^(1/2)/(e*x^2+d)+1/4*x 
^4*(a+b*arccsch(c*x))/d/(e*x^2+d)^2+1/8*b*c*(c^2*d-2*e)*x*arctanh(e^(1/2)* 
(-c^2*x^2-1)^(1/2)/(c^2*d-e)^(1/2))/d/(c^2*d-e)^(3/2)/e^(3/2)/(-c^2*x^2)^( 
1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.87 (sec) , antiderivative size = 375, normalized size of antiderivative = 2.25 \[ \int \frac {x^3 \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=-\frac {-\frac {4 a d}{\left (d+e x^2\right )^2}+\frac {8 a}{d+e x^2}-\frac {2 b c e \sqrt {1+\frac {1}{c^2 x^2}} x}{\left (-c^2 d+e\right ) \left (d+e x^2\right )}+\frac {4 b \left (d+2 e x^2\right ) \text {csch}^{-1}(c x)}{\left (d+e x^2\right )^2}-\frac {4 b \text {arcsinh}\left (\frac {1}{c x}\right )}{d}+\frac {b \sqrt {e} \left (-c^2 d+2 e\right ) \log \left (\frac {16 d e^{3/2} \sqrt {-c^2 d+e} \left (\sqrt {e}+c \left (-i c \sqrt {d}+\sqrt {-c^2 d+e} \sqrt {1+\frac {1}{c^2 x^2}}\right ) x\right )}{b \left (-c^2 d+2 e\right ) \left (i \sqrt {d}+\sqrt {e} x\right )}\right )}{d \left (-c^2 d+e\right )^{3/2}}+\frac {b \sqrt {e} \left (-c^2 d+2 e\right ) \log \left (-\frac {16 i d e^{3/2} \sqrt {-c^2 d+e} \left (\sqrt {e}+c \left (i c \sqrt {d}+\sqrt {-c^2 d+e} \sqrt {1+\frac {1}{c^2 x^2}}\right ) x\right )}{b \left (c^2 d-2 e\right ) \left (\sqrt {d}+i \sqrt {e} x\right )}\right )}{d \left (-c^2 d+e\right )^{3/2}}}{16 e^2} \] Input:

Integrate[(x^3*(a + b*ArcCsch[c*x]))/(d + e*x^2)^3,x]
 

Output:

-1/16*((-4*a*d)/(d + e*x^2)^2 + (8*a)/(d + e*x^2) - (2*b*c*e*Sqrt[1 + 1/(c 
^2*x^2)]*x)/((-(c^2*d) + e)*(d + e*x^2)) + (4*b*(d + 2*e*x^2)*ArcCsch[c*x] 
)/(d + e*x^2)^2 - (4*b*ArcSinh[1/(c*x)])/d + (b*Sqrt[e]*(-(c^2*d) + 2*e)*L 
og[(16*d*e^(3/2)*Sqrt[-(c^2*d) + e]*(Sqrt[e] + c*((-I)*c*Sqrt[d] + Sqrt[-( 
c^2*d) + e]*Sqrt[1 + 1/(c^2*x^2)])*x))/(b*(-(c^2*d) + 2*e)*(I*Sqrt[d] + Sq 
rt[e]*x))])/(d*(-(c^2*d) + e)^(3/2)) + (b*Sqrt[e]*(-(c^2*d) + 2*e)*Log[((- 
16*I)*d*e^(3/2)*Sqrt[-(c^2*d) + e]*(Sqrt[e] + c*(I*c*Sqrt[d] + Sqrt[-(c^2* 
d) + e]*Sqrt[1 + 1/(c^2*x^2)])*x))/(b*(c^2*d - 2*e)*(Sqrt[d] + I*Sqrt[e]*x 
))])/(d*(-(c^2*d) + e)^(3/2)))/e^2
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.92, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6856, 27, 354, 87, 73, 221}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {x^3 \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx\)

\(\Big \downarrow \) 6856

\(\displaystyle \frac {x^4 \left (a+b \text {csch}^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}-\frac {b c x \int \frac {x^3}{4 d \sqrt {-c^2 x^2-1} \left (e x^2+d\right )^2}dx}{\sqrt {-c^2 x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x^4 \left (a+b \text {csch}^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}-\frac {b c x \int \frac {x^3}{\sqrt {-c^2 x^2-1} \left (e x^2+d\right )^2}dx}{4 d \sqrt {-c^2 x^2}}\)

\(\Big \downarrow \) 354

\(\displaystyle \frac {x^4 \left (a+b \text {csch}^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}-\frac {b c x \int \frac {x^2}{\sqrt {-c^2 x^2-1} \left (e x^2+d\right )^2}dx^2}{8 d \sqrt {-c^2 x^2}}\)

\(\Big \downarrow \) 87

\(\displaystyle \frac {x^4 \left (a+b \text {csch}^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}-\frac {b c x \left (\frac {\left (c^2 d-2 e\right ) \int \frac {1}{\sqrt {-c^2 x^2-1} \left (e x^2+d\right )}dx^2}{2 e \left (c^2 d-e\right )}+\frac {d \sqrt {-c^2 x^2-1}}{e \left (c^2 d-e\right ) \left (d+e x^2\right )}\right )}{8 d \sqrt {-c^2 x^2}}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {x^4 \left (a+b \text {csch}^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}-\frac {b c x \left (\frac {d \sqrt {-c^2 x^2-1}}{e \left (c^2 d-e\right ) \left (d+e x^2\right )}-\frac {\left (c^2 d-2 e\right ) \int \frac {1}{-\frac {e x^4}{c^2}+d-\frac {e}{c^2}}d\sqrt {-c^2 x^2-1}}{c^2 e \left (c^2 d-e\right )}\right )}{8 d \sqrt {-c^2 x^2}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {x^4 \left (a+b \text {csch}^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}-\frac {b c x \left (\frac {d \sqrt {-c^2 x^2-1}}{e \left (c^2 d-e\right ) \left (d+e x^2\right )}-\frac {\left (c^2 d-2 e\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {-c^2 x^2-1}}{\sqrt {c^2 d-e}}\right )}{e^{3/2} \left (c^2 d-e\right )^{3/2}}\right )}{8 d \sqrt {-c^2 x^2}}\)

Input:

Int[(x^3*(a + b*ArcCsch[c*x]))/(d + e*x^2)^3,x]
 

Output:

(x^4*(a + b*ArcCsch[c*x]))/(4*d*(d + e*x^2)^2) - (b*c*x*((d*Sqrt[-1 - c^2* 
x^2])/((c^2*d - e)*e*(d + e*x^2)) - ((c^2*d - 2*e)*ArcTanh[(Sqrt[e]*Sqrt[- 
1 - c^2*x^2])/Sqrt[c^2*d - e]])/((c^2*d - e)^(3/2)*e^(3/2))))/(8*d*Sqrt[-( 
c^2*x^2)])
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 87

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] 
/; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege 
rQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ[p, n]))))
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 354

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S 
ymbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x 
, x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ 
[(m - 1)/2]
 

rule 6856

Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*( 
x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Si 
mp[(a + b*ArcCsch[c*x])   u, x] - Simp[b*c*(x/Sqrt[(-c^2)*x^2])   Int[Simpl 
ifyIntegrand[u/(x*Sqrt[-1 - c^2*x^2]), x], x], x]] /; FreeQ[{a, b, c, d, e, 
 f, m, p}, x] && ((IGtQ[p, 0] &&  !(ILtQ[(m - 1)/2, 0] && GtQ[m + 2*p + 3, 
0])) || (IGtQ[(m + 1)/2, 0] &&  !(ILtQ[p, 0] && GtQ[m + 2*p + 3, 0])) || (I 
LtQ[(m + 2*p + 1)/2, 0] &&  !ILtQ[(m - 1)/2, 0]))
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(962\) vs. \(2(145)=290\).

Time = 6.16 (sec) , antiderivative size = 963, normalized size of antiderivative = 5.77

method	result	size
parts	\(a \left (-\frac {1}{2 e^{2} \left (x^{2} e +d \right )}+\frac {d}{4 e^{2} \left (x^{2} e +d \right )^{2}}\right )+\frac {b \left (\frac {c^{8} \operatorname {arccsch}\left (c x \right ) d}{4 e^{2} \left (e \,c^{2} x^{2}+c^{2} d \right )^{2}}-\frac {c^{6} \operatorname {arccsch}\left (c x \right )}{2 e^{2} \left (e \,c^{2} x^{2}+c^{2} d \right )}+\frac {c^{3} \sqrt {c^{2} x^{2}+1}\, \left (-4 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right ) \sqrt {-\frac {c^{2} d -e}{e}}\, c^{4} d^{2}-4 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right ) \sqrt {-\frac {c^{2} d -e}{e}}\, c^{4} d e \,x^{2}+\ln \left (-\frac {2 \left (\sqrt {-\frac {c^{2} d -e}{e}}\, \sqrt {c^{2} x^{2}+1}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) c^{4} d^{2}+\ln \left (-\frac {2 \left (\sqrt {-\frac {c^{2} d -e}{e}}\, \sqrt {c^{2} x^{2}+1}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) c^{4} d e \,x^{2}+\ln \left (-\frac {2 \left (-\sqrt {-\frac {c^{2} d -e}{e}}\, \sqrt {c^{2} x^{2}+1}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{c e x +\sqrt {-c^{2} d e}}\right ) c^{4} d^{2}+\ln \left (-\frac {2 \left (-\sqrt {-\frac {c^{2} d -e}{e}}\, \sqrt {c^{2} x^{2}+1}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{c e x +\sqrt {-c^{2} d e}}\right ) c^{4} d e \,x^{2}+2 \sqrt {c^{2} x^{2}+1}\, \sqrt {-\frac {c^{2} d -e}{e}}\, c^{2} d e +4 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right ) \sqrt {-\frac {c^{2} d -e}{e}}\, c^{2} d e +4 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right ) \sqrt {-\frac {c^{2} d -e}{e}}\, e^{2} c^{2} x^{2}-2 \ln \left (-\frac {2 \left (\sqrt {-\frac {c^{2} d -e}{e}}\, \sqrt {c^{2} x^{2}+1}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) c^{2} d e -2 \ln \left (-\frac {2 \left (\sqrt {-\frac {c^{2} d -e}{e}}\, \sqrt {c^{2} x^{2}+1}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) e^{2} c^{2} x^{2}-2 \ln \left (-\frac {2 \left (-\sqrt {-\frac {c^{2} d -e}{e}}\, \sqrt {c^{2} x^{2}+1}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{c e x +\sqrt {-c^{2} d e}}\right ) c^{2} d e -2 \ln \left (-\frac {2 \left (-\sqrt {-\frac {c^{2} d -e}{e}}\, \sqrt {c^{2} x^{2}+1}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{c e x +\sqrt {-c^{2} d e}}\right ) e^{2} c^{2} x^{2}\right )}{16 e \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, x d \left (c^{2} d -e \right ) \left (-c e x +\sqrt {-c^{2} d e}\right ) \sqrt {-\frac {c^{2} d -e}{e}}\, \left (c e x +\sqrt {-c^{2} d e}\right )}\right )}{c^{4}}\)	\(963\)
derivativedivides	\(\frac {a \,c^{6} \left (-\frac {1}{2 e^{2} \left (e \,c^{2} x^{2}+c^{2} d \right )}+\frac {d \,c^{2}}{4 e^{2} \left (e \,c^{2} x^{2}+c^{2} d \right )^{2}}\right )+b \,c^{6} \left (-\frac {\operatorname {arccsch}\left (c x \right )}{2 e^{2} \left (e \,c^{2} x^{2}+c^{2} d \right )}+\frac {\operatorname {arccsch}\left (c x \right ) d \,c^{2}}{4 e^{2} \left (e \,c^{2} x^{2}+c^{2} d \right )^{2}}+\frac {\sqrt {c^{2} x^{2}+1}\, \left (-4 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right ) \sqrt {-\frac {c^{2} d -e}{e}}\, c^{4} d^{2}-4 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right ) \sqrt {-\frac {c^{2} d -e}{e}}\, c^{4} d e \,x^{2}+\ln \left (-\frac {2 \left (\sqrt {-\frac {c^{2} d -e}{e}}\, \sqrt {c^{2} x^{2}+1}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) c^{4} d^{2}+\ln \left (-\frac {2 \left (\sqrt {-\frac {c^{2} d -e}{e}}\, \sqrt {c^{2} x^{2}+1}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) c^{4} d e \,x^{2}+\ln \left (-\frac {2 \left (-\sqrt {-\frac {c^{2} d -e}{e}}\, \sqrt {c^{2} x^{2}+1}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{c e x +\sqrt {-c^{2} d e}}\right ) c^{4} d^{2}+\ln \left (-\frac {2 \left (-\sqrt {-\frac {c^{2} d -e}{e}}\, \sqrt {c^{2} x^{2}+1}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{c e x +\sqrt {-c^{2} d e}}\right ) c^{4} d e \,x^{2}+2 \sqrt {c^{2} x^{2}+1}\, \sqrt {-\frac {c^{2} d -e}{e}}\, c^{2} d e +4 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right ) \sqrt {-\frac {c^{2} d -e}{e}}\, c^{2} d e +4 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right ) \sqrt {-\frac {c^{2} d -e}{e}}\, e^{2} c^{2} x^{2}-2 \ln \left (-\frac {2 \left (\sqrt {-\frac {c^{2} d -e}{e}}\, \sqrt {c^{2} x^{2}+1}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) c^{2} d e -2 \ln \left (-\frac {2 \left (\sqrt {-\frac {c^{2} d -e}{e}}\, \sqrt {c^{2} x^{2}+1}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) e^{2} c^{2} x^{2}-2 \ln \left (-\frac {2 \left (-\sqrt {-\frac {c^{2} d -e}{e}}\, \sqrt {c^{2} x^{2}+1}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{c e x +\sqrt {-c^{2} d e}}\right ) c^{2} d e -2 \ln \left (-\frac {2 \left (-\sqrt {-\frac {c^{2} d -e}{e}}\, \sqrt {c^{2} x^{2}+1}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{c e x +\sqrt {-c^{2} d e}}\right ) e^{2} c^{2} x^{2}\right )}{16 e \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c^{3} x d \left (c^{2} d -e \right ) \left (-c e x +\sqrt {-c^{2} d e}\right ) \sqrt {-\frac {c^{2} d -e}{e}}\, \left (c e x +\sqrt {-c^{2} d e}\right )}\right )}{c^{4}}\)	\(984\)
default	\(\frac {a \,c^{6} \left (-\frac {1}{2 e^{2} \left (e \,c^{2} x^{2}+c^{2} d \right )}+\frac {d \,c^{2}}{4 e^{2} \left (e \,c^{2} x^{2}+c^{2} d \right )^{2}}\right )+b \,c^{6} \left (-\frac {\operatorname {arccsch}\left (c x \right )}{2 e^{2} \left (e \,c^{2} x^{2}+c^{2} d \right )}+\frac {\operatorname {arccsch}\left (c x \right ) d \,c^{2}}{4 e^{2} \left (e \,c^{2} x^{2}+c^{2} d \right )^{2}}+\frac {\sqrt {c^{2} x^{2}+1}\, \left (-4 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right ) \sqrt {-\frac {c^{2} d -e}{e}}\, c^{4} d^{2}-4 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right ) \sqrt {-\frac {c^{2} d -e}{e}}\, c^{4} d e \,x^{2}+\ln \left (-\frac {2 \left (\sqrt {-\frac {c^{2} d -e}{e}}\, \sqrt {c^{2} x^{2}+1}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) c^{4} d^{2}+\ln \left (-\frac {2 \left (\sqrt {-\frac {c^{2} d -e}{e}}\, \sqrt {c^{2} x^{2}+1}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) c^{4} d e \,x^{2}+\ln \left (-\frac {2 \left (-\sqrt {-\frac {c^{2} d -e}{e}}\, \sqrt {c^{2} x^{2}+1}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{c e x +\sqrt {-c^{2} d e}}\right ) c^{4} d^{2}+\ln \left (-\frac {2 \left (-\sqrt {-\frac {c^{2} d -e}{e}}\, \sqrt {c^{2} x^{2}+1}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{c e x +\sqrt {-c^{2} d e}}\right ) c^{4} d e \,x^{2}+2 \sqrt {c^{2} x^{2}+1}\, \sqrt {-\frac {c^{2} d -e}{e}}\, c^{2} d e +4 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right ) \sqrt {-\frac {c^{2} d -e}{e}}\, c^{2} d e +4 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right ) \sqrt {-\frac {c^{2} d -e}{e}}\, e^{2} c^{2} x^{2}-2 \ln \left (-\frac {2 \left (\sqrt {-\frac {c^{2} d -e}{e}}\, \sqrt {c^{2} x^{2}+1}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) c^{2} d e -2 \ln \left (-\frac {2 \left (\sqrt {-\frac {c^{2} d -e}{e}}\, \sqrt {c^{2} x^{2}+1}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) e^{2} c^{2} x^{2}-2 \ln \left (-\frac {2 \left (-\sqrt {-\frac {c^{2} d -e}{e}}\, \sqrt {c^{2} x^{2}+1}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{c e x +\sqrt {-c^{2} d e}}\right ) c^{2} d e -2 \ln \left (-\frac {2 \left (-\sqrt {-\frac {c^{2} d -e}{e}}\, \sqrt {c^{2} x^{2}+1}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{c e x +\sqrt {-c^{2} d e}}\right ) e^{2} c^{2} x^{2}\right )}{16 e \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c^{3} x d \left (c^{2} d -e \right ) \left (-c e x +\sqrt {-c^{2} d e}\right ) \sqrt {-\frac {c^{2} d -e}{e}}\, \left (c e x +\sqrt {-c^{2} d e}\right )}\right )}{c^{4}}\)	\(984\)

Input:

int(x^3*(a+b*arccsch(c*x))/(e*x^2+d)^3,x,method=_RETURNVERBOSE)
 

Output:

a*(-1/2/e^2/(e*x^2+d)+1/4/e^2*d/(e*x^2+d)^2)+b/c^4*(1/4*c^8*arccsch(c*x)*d 
/e^2/(c^2*e*x^2+c^2*d)^2-1/2*c^6*arccsch(c*x)/e^2/(c^2*e*x^2+c^2*d)+1/16*c 
^3*(c^2*x^2+1)^(1/2)/e*(-4*arctanh(1/(c^2*x^2+1)^(1/2))*(-(c^2*d-e)/e)^(1/ 
2)*c^4*d^2-4*arctanh(1/(c^2*x^2+1)^(1/2))*(-(c^2*d-e)/e)^(1/2)*c^4*d*e*x^2 
+ln(-2*((-(c^2*d-e)/e)^(1/2)*(c^2*x^2+1)^(1/2)*e+(-c^2*d*e)^(1/2)*c*x+e)/( 
-c*e*x+(-c^2*d*e)^(1/2)))*c^4*d^2+ln(-2*((-(c^2*d-e)/e)^(1/2)*(c^2*x^2+1)^ 
(1/2)*e+(-c^2*d*e)^(1/2)*c*x+e)/(-c*e*x+(-c^2*d*e)^(1/2)))*c^4*d*e*x^2+ln( 
-2*(-(-(c^2*d-e)/e)^(1/2)*(c^2*x^2+1)^(1/2)*e+(-c^2*d*e)^(1/2)*c*x-e)/(c*e 
*x+(-c^2*d*e)^(1/2)))*c^4*d^2+ln(-2*(-(-(c^2*d-e)/e)^(1/2)*(c^2*x^2+1)^(1/ 
2)*e+(-c^2*d*e)^(1/2)*c*x-e)/(c*e*x+(-c^2*d*e)^(1/2)))*c^4*d*e*x^2+2*(c^2* 
x^2+1)^(1/2)*(-(c^2*d-e)/e)^(1/2)*c^2*d*e+4*arctanh(1/(c^2*x^2+1)^(1/2))*( 
-(c^2*d-e)/e)^(1/2)*c^2*d*e+4*arctanh(1/(c^2*x^2+1)^(1/2))*(-(c^2*d-e)/e)^ 
(1/2)*e^2*c^2*x^2-2*ln(-2*((-(c^2*d-e)/e)^(1/2)*(c^2*x^2+1)^(1/2)*e+(-c^2* 
d*e)^(1/2)*c*x+e)/(-c*e*x+(-c^2*d*e)^(1/2)))*c^2*d*e-2*ln(-2*((-(c^2*d-e)/ 
e)^(1/2)*(c^2*x^2+1)^(1/2)*e+(-c^2*d*e)^(1/2)*c*x+e)/(-c*e*x+(-c^2*d*e)^(1 
/2)))*e^2*c^2*x^2-2*ln(-2*(-(-(c^2*d-e)/e)^(1/2)*(c^2*x^2+1)^(1/2)*e+(-c^2 
*d*e)^(1/2)*c*x-e)/(c*e*x+(-c^2*d*e)^(1/2)))*c^2*d*e-2*ln(-2*(-(-(c^2*d-e) 
/e)^(1/2)*(c^2*x^2+1)^(1/2)*e+(-c^2*d*e)^(1/2)*c*x-e)/(c*e*x+(-c^2*d*e)^(1 
/2)))*e^2*c^2*x^2)/((c^2*x^2+1)/c^2/x^2)^(1/2)/x/d/(c^2*d-e)/(-c*e*x+(-c^2 
*d*e)^(1/2))/(-(c^2*d-e)/e)^(1/2)/(c*e*x+(-c^2*d*e)^(1/2)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 682 vs. \(2 (145) = 290\).

Time = 0.29 (sec) , antiderivative size = 1381, normalized size of antiderivative = 8.27 \[ \int \frac {x^3 \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(x^3*(a+b*arccsch(c*x))/(e*x^2+d)^3,x, algorithm="fricas")
 

Output:

[-1/16*(4*a*c^4*d^4 - 8*a*c^2*d^3*e + 4*a*d^2*e^2 + 8*(a*c^4*d^3*e - 2*a*c 
^2*d^2*e^2 + a*d*e^3)*x^2 + (b*c^2*d^3 + (b*c^2*d*e^2 - 2*b*e^3)*x^4 - 2*b 
*d^2*e + 2*(b*c^2*d^2*e - 2*b*d*e^2)*x^2)*sqrt(-c^2*d*e + e^2)*log((c^2*e* 
x^2 - c^2*d - 2*sqrt(-c^2*d*e + e^2)*c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 2 
*e)/(e*x^2 + d)) - 4*(b*c^4*d^4 - 2*b*c^2*d^3*e + b*d^2*e^2 + (b*c^4*d^2*e 
^2 - 2*b*c^2*d*e^3 + b*e^4)*x^4 + 2*(b*c^4*d^3*e - 2*b*c^2*d^2*e^2 + b*d*e 
^3)*x^2)*log(c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) - c*x + 1) + 4*(b*c^4*d^4 - 
 2*b*c^2*d^3*e + b*d^2*e^2 + (b*c^4*d^2*e^2 - 2*b*c^2*d*e^3 + b*e^4)*x^4 + 
 2*(b*c^4*d^3*e - 2*b*c^2*d^2*e^2 + b*d*e^3)*x^2)*log(c*x*sqrt((c^2*x^2 + 
1)/(c^2*x^2)) - c*x - 1) + 4*(b*c^4*d^4 - 2*b*c^2*d^3*e + b*d^2*e^2 + 2*(b 
*c^4*d^3*e - 2*b*c^2*d^2*e^2 + b*d*e^3)*x^2)*log((c*x*sqrt((c^2*x^2 + 1)/( 
c^2*x^2)) + 1)/(c*x)) + 2*((b*c^3*d^2*e^2 - b*c*d*e^3)*x^3 + (b*c^3*d^3*e 
- b*c*d^2*e^2)*x)*sqrt((c^2*x^2 + 1)/(c^2*x^2)))/(c^4*d^5*e^2 - 2*c^2*d^4* 
e^3 + d^3*e^4 + (c^4*d^3*e^4 - 2*c^2*d^2*e^5 + d*e^6)*x^4 + 2*(c^4*d^4*e^3 
 - 2*c^2*d^3*e^4 + d^2*e^5)*x^2), -1/8*(2*a*c^4*d^4 - 4*a*c^2*d^3*e + 2*a* 
d^2*e^2 + 4*(a*c^4*d^3*e - 2*a*c^2*d^2*e^2 + a*d*e^3)*x^2 + (b*c^2*d^3 + ( 
b*c^2*d*e^2 - 2*b*e^3)*x^4 - 2*b*d^2*e + 2*(b*c^2*d^2*e - 2*b*d*e^2)*x^2)* 
sqrt(c^2*d*e - e^2)*arctan(sqrt(c^2*d*e - e^2)*c*x*sqrt((c^2*x^2 + 1)/(c^2 
*x^2))/(c^2*e*x^2 + e)) - 2*(b*c^4*d^4 - 2*b*c^2*d^3*e + b*d^2*e^2 + (b*c^ 
4*d^2*e^2 - 2*b*c^2*d*e^3 + b*e^4)*x^4 + 2*(b*c^4*d^3*e - 2*b*c^2*d^2*e...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^3 \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate(x**3*(a+b*acsch(c*x))/(e*x**2+d)**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {x^3 \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x^{3}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \] Input:

integrate(x^3*(a+b*arccsch(c*x))/(e*x^2+d)^3,x, algorithm="maxima")
                                                                                    
                                                                                    
 

Output:

1/8*b*((2*c^4*d^4*log(c) - 2*(c^4*d^2*e^2 - 2*c^2*d*e^3 + e^4)*x^4*log(x) 
+ 2*d^2*e^2*log(c) + d^2*e^2 - (4*d^3*e*log(c) + d^3*e)*c^2 + (4*c^4*d^3*e 
*log(c) + 4*d*e^3*log(c) + d*e^3 - (8*d^2*e^2*log(c) + d^2*e^2)*c^2)*x^2 + 
 (c^4*d^4 - 2*c^2*d^3*e + (c^4*d^2*e^2 - 2*c^2*d*e^3)*x^4 + 2*(c^4*d^3*e - 
 2*c^2*d^2*e^2)*x^2)*log(c^2*x^2 + 1) - 2*(c^4*d^4 - 2*c^2*d^3*e + d^2*e^2 
 + 2*(c^4*d^3*e - 2*c^2*d^2*e^2 + d*e^3)*x^2)*log(sqrt(c^2*x^2 + 1) + 1))/ 
(c^4*d^5*e^2 - 2*c^2*d^4*e^3 + d^3*e^4 + (c^4*d^3*e^4 - 2*c^2*d^2*e^5 + d* 
e^6)*x^4 + 2*(c^4*d^4*e^3 - 2*c^2*d^3*e^4 + d^2*e^5)*x^2) + log(e*x^2 + d) 
/(c^4*d^3 - 2*c^2*d^2*e + d*e^2) - 8*integrate(1/4*(2*c^2*e*x^3 + c^2*d*x) 
/(c^2*e^4*x^6 + (2*c^2*d*e^3 + e^4)*x^4 + d^2*e^2 + (c^2*d^2*e^2 + 2*d*e^3 
)*x^2 + (c^2*e^4*x^6 + (2*c^2*d*e^3 + e^4)*x^4 + d^2*e^2 + (c^2*d^2*e^2 + 
2*d*e^3)*x^2)*sqrt(c^2*x^2 + 1)), x)) - 1/4*(2*e*x^2 + d)*a/(e^4*x^4 + 2*d 
*e^3*x^2 + d^2*e^2)
 

Giac [F]

\[ \int \frac {x^3 \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x^{3}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \] Input:

integrate(x^3*(a+b*arccsch(c*x))/(e*x^2+d)^3,x, algorithm="giac")
 

Output:

integrate((b*arccsch(c*x) + a)*x^3/(e*x^2 + d)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\int \frac {x^3\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^3} \,d x \] Input:

int((x^3*(a + b*asinh(1/(c*x))))/(d + e*x^2)^3,x)
 

Output:

int((x^3*(a + b*asinh(1/(c*x))))/(d + e*x^2)^3, x)
 

Reduce [F]

\[ \int \frac {x^3 \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\frac {4 \left (\int \frac {\mathit {acsch} \left (c x \right ) x^{3}}{e^{3} x^{6}+3 d \,e^{2} x^{4}+3 d^{2} e \,x^{2}+d^{3}}d x \right ) b \,d^{3}+8 \left (\int \frac {\mathit {acsch} \left (c x \right ) x^{3}}{e^{3} x^{6}+3 d \,e^{2} x^{4}+3 d^{2} e \,x^{2}+d^{3}}d x \right ) b \,d^{2} e \,x^{2}+4 \left (\int \frac {\mathit {acsch} \left (c x \right ) x^{3}}{e^{3} x^{6}+3 d \,e^{2} x^{4}+3 d^{2} e \,x^{2}+d^{3}}d x \right ) b d \,e^{2} x^{4}+a \,x^{4}}{4 d \left (e^{2} x^{4}+2 d e \,x^{2}+d^{2}\right )} \] Input:

int(x^3*(a+b*acsch(c*x))/(e*x^2+d)^3,x)
 

Output:

(4*int((acsch(c*x)*x**3)/(d**3 + 3*d**2*e*x**2 + 3*d*e**2*x**4 + e**3*x**6 
),x)*b*d**3 + 8*int((acsch(c*x)*x**3)/(d**3 + 3*d**2*e*x**2 + 3*d*e**2*x** 
4 + e**3*x**6),x)*b*d**2*e*x**2 + 4*int((acsch(c*x)*x**3)/(d**3 + 3*d**2*e 
*x**2 + 3*d*e**2*x**4 + e**3*x**6),x)*b*d*e**2*x**4 + a*x**4)/(4*d*(d**2 + 
 2*d*e*x**2 + e**2*x**4))